Symplectic homology of convex domains and Clarke's duality

TL;DR

This paper establishes an isomorphism between Floer and Morse complexes for convex Hamiltonians, linking symplectic homology to boundary characteristics, and showing the equivalence of symplectic capacity and minimal boundary action.

Contribution

It proves a novel isomorphism between Floer and Morse complexes in convex domains, connecting symplectic homology with boundary dynamics.

Findings

Floer complex is isomorphic to Morse complex of Clarke's dual functional

Symplectic capacity equals minimal action of boundary closed characteristics

Isomorphism preserves action filtrations

Abstract

We prove that the Floer complex that is associated with a convex Hamiltonian function on $\mathbb{R}^{2n}$ is isomorphic to the Morse complex of Clarke's dual action functional that is associated with the Fenchel-dual Hamiltonian. This isomorphism preserves the action filtrations. As a corollary, we obtain that the symplectic capacity from the symplectic homology of a convex domain with smooth boundary coincides with the minimal action of closed characteristics on its boundary.